The parameterized complexity of k-edge induced subgraphs ⋆ Bingkai Lin1 and Yijia Chen1 Shanghai Jiao Tong University [email protected], [email protected]

Abstract. We prove that finding a k-edge induced subgraph is fixedparameter tractable, thereby answering an open problem of Leizhen Cai [2]. Our algorithm is based on several combinatorial observations, Gauss’ famous Eureka theorem [1], and a generalization of the wellknown fpt-algorithm for the model-checking problem for first-order logic on graphs with locally bounded tree-width due to Frick and Grohe [13]. On the other hand, we show that two natural counting versions of the problem are hard. Hence, the k-edge induced subgraph problem is one of the very few known examples in parameterized complexity that are easy for decision while hard for counting.

1

Introduction

Induced subgraphs are one of the most natural substructures in graphs. They capture many different combinatorial objects, e.g., clique, independent set, chordless path. Thus, a great number of algorithmic problems are about finding certain induced subgraphs, and their complexity is among the mostly extensively studied in algorithmic graph theory [3, 7, 14]. In this paper, we are mainly interested in the problem of finding an induced subgraph which contains exactly k edges, i.e., a k-edge induced subgraph. This problem is equivalent to solving a special 0-1 quadratic Diophantine equation xT Ax = k, where A is the adjacent matrix of G, x ∈ {0, 1}n, n = |V (G)|. It is not difficult to prove that the k-edge induced subgraph problem is NPhard by a reduction from the clique problem. So we approach the problem via parameterized complexity [9, 12] and treat k as the parameter: p-Edge-Induced-Subgraph Instance: A graph G and k ∈ N. Parameter: k. Problem: Decide whether G contains a k-edge induced subgraph. As the main result of our paper, we show that p-Edge-Induced-Subgraph is fixed-parameter tractable. In fact, there are special cases of p-Edge-InducedSubgraph whose fixed-parameter tractability has been known for a while. Since ⋆

Full version available at http://arxiv.org/abs/1105.0477

we can define a k-edge induced subgraph by a first-order sentence, using logic machinery, it can be shown that p-Edge-Induced-Subgraph is fixed-parameter tractable if the graph G has bounded tree-width [8], bounded local tree-width [13], etc., or most generally locally bounded expansion [10]. Unfortunately, the class of all graphs containing a k-edge induced subgraph does not possess any of these bounded measures. As another previously known case, using his Random Separation method [5] and Ramsey’s Theorem, Cai [4] gave a very nice combinatorial algorithm that solves p-Edge-Induced-Subgraph when the parameter k is a  triangular number, i.e., k = m for some m ∈ N. However, it looks very diffi2 cult to adapt Cai’s algorithm to handle arbitrary k. Therefore neither logic nor combinatorial approach so far seems to be sufficient to settle the complexity of p-Edge-Induced-Subgraph by its own. So our fpt-algorithm is a rather tricky combination of these two methods. 1.1

Our approach

As just mentioned, our starting pointing is that the existence of a k-edge induced subgraph can be characterized by a sentence of first-order logic (FO) which depends on k only. It is a well-known result of Frick and Grohe [13] that the model-checking problem for FO on graphs of bounded local tree-width is fixedparameter tractable. The local tree-width for a graph is a function bounding the tree-width of the induced subgraphs on the neighborhoods within a certain radius of every vertex. For instance, bounded-degree graphs have bounded local tree-width. These give immediately the fixed-parameter tractability of p-Edge-Induced-Subgraph on graphs with bounded degree1 . With some more efforts, the above result can be extended to graphs G with degree bounded by a function of the parameter k. In that case, we can say the degree deg(v) of each vertex v is sufficiently small. The corresponding fptalgorithm generalizes Frick and Grohe’s Theorem to graphs with local tree-width bounded by a function of both the radius of the neighborhoods and an additional parameter. As a dual, if deg(v) of each vertex v in G is sufficiently large, or more precisely, the complement of G has degree bounded by a function of k, then we can decide p-Edge-Induced-Subgraph in fpt time, too. Moving one step further, we consider graphs in which each deg(v) is either sufficiently small or sufficiently large, e.g., an n-star. We call such graphs degreeextreme. Using the same logic machinery as above, we are able to show the fixed-parameter tractability of p-Edge-Induced-Subgraph on degree-extreme graphs. Assume that the graph G is not degree-extreme, i.e., there exists a vertex v0 whose degree is neither sufficiently small nor sufficiently large. We partition the vertex set of G into two sets V1 and V2 , where V1 contains all vertices adjacent to v0 and V2 the remaining vertices. Then both V1 and V2 are relatively large. Note 1

This is also a direct consequence of Seese’s result that the model-checking problem for FO on bounded-degree graphs is fixed-parameter tractable [15]. But we find it more natural to work with bounded local tree-with in the following generalization.

possibly there are many edges between V1 and V2 . Nevertheless, we can compute a vertex set B in G such that every edge between V1 and V2 has one vertex in B; and if B is large enough, we can show that G contains a k-edge induced subgraph. Otherwise, the graph G consists of two induced subgraphs G[V1 ] and G[V2 ], plus the edges between V1 and V2 adjacent to the set B of bounded size. In case G[V1 ] and G[V2 ] are both degree-extreme, we call such a graph G a bridge (of two degree-extreme graphs). By the logic method again, we prove that p-Edge-Induced-Subgraph is fixed-parameter tractable on bridges. Now we are left with the case that at least one of G[V1 ] and G[V2 ] is not degree-extreme, say G[V1 ]. Then we repeat the above procedure on G[V1 ] to get a partition V11 | V12 of V1 . And again, both V11 and V12 are sufficiently large. Arguing as before, either we already know G[V1 ], and hence G, contains a kedge induced subgraph, or there is a set B1 of bounded size such that every edge between V11 and V22 intersects B1 . Finally we remove the vertex set B0 := B ∪ B1 from G. Then G[V \ B0 ] is the disjoint union of G[V11 \ B0 ], G[V12 \ B0 ] and G[V2 \ B0 ]. Moreover, all the three induced subgraphs are so large that, by Ramsey’s Theorem, either one of them contains a large independent set, or we have three large disjoint cliques which are not adjacent to each other. For both cases, we show that G[V \ B0 ], and hence G, contains a k-edge induced subgraph. As a matter of fact, the second case is an easy consequence of a famous number-theoretic result of Gauss which states that every natural number is the sum of three triangular numbers. We should mention that the running time of our algorithm in terms of the parameter k is triple exponential at least. On the other hand, it is linear in the size of the graph. We leave the detailed analysis in the full version of the paper. 1.2

Counting k-edge induced subgraphs

We also study the parameterized complexity of computing the number of kedge induced subgraphs. For most natural problems, if the decision version is easy, then so is the counting problem. However, it turns out that two natural counting versions of p-Edge-Induced-Subgraph are both hard. To the best of our knowledge, there are only few natural problems which exhibit such a phenomenon [11, 6]. 1.3

Organization of our paper

In Section 2 we introduce necessary background and fix our notations. We prove all required combinatorial results in Section 3. In particular, we present several simple structures in a graph which, if exist, guarantee the existence of a k-edge induced subgraph. Then in Section 4 we establish the fixed-parameter tractability of p-Edge-Induced-Subgraph on the degree-extreme graphs and the bridges using model-checking problems for FO. We present our fpt-algorithm for p-Edge-Induced-Subgraph by putting all the pieces together in Section 5. Finally in Section 6 we prove the hardness of the counting problems. Due to the space limitations, for some proofs we refer to the full version of this paper.

2

Preliminaries

N and N+ denote the sets of natural numbers (that is, nonnegative integers) and positive integers, respectively. For a natural number n let [n] := {1, . . . , n}. We denote the alphabet {0, 1} by Σ and identify problems with subsets Q of Σ ∗ . Clearly, as done mostly, we present concrete problems in a verbal, hence uncodified its size. Moreover we let  form over Σ. For every set S we use |S| to denote  S subsets of S, i.e., {a, b} a, b ∈ S and a 6= b . 2 be the set of all two-element     for some k ∈ N. In particular, 0 = 1 = 0. A triangular number is k2 := [k] 2 2 2 2.1

Parameterized complexity

A parameterized problem is a pair (Q, κ) consisting of a classical problem Q ⊆ Σ ∗ and a polynomial time computable parameterization κ : Σ ∗ → N. An algorithm A is an fpt-algorithm with respect to a parameterization κ if for every x ∈ Σ ∗ the running time of A on x is bounded by f (κ(x)) · |x|O(1) for a computable function f : N → N. Or equivalently, we say that the algorithm A runs in fpt time. A parameterized problem (Q, κ) is fixed-parameter tractable if there is an fpt-algorithm with respect to κ that decides Q. 2.2

Graphs

We only consider simple graphs, that is, finite nonempty undirected graphs without loops and parallel edges. Every graph G = (V,  E) is thus determined by a nonempty vertex set V and an edge set E ⊆ V2 . For an edge {u, v} ∈ E we say that u is adjacent to v, and vice versa. Often we also use V (G) and E(G) to denote the vertex set and the edge set of G, respectively. G Let G = (V, E) be a graph. For every vertex v ∈ V the set N (v) contains all G vertices in G that are adjacent to v,Si.e., N (v) := u {u, v} ∈ E . Moreover, for every S ⊆ V we let N G (S) := v∈S N G (v). Note the degree of v, written degG (v), is |N G (v)|. If degG (v) = 0, then v is an isolated vertex. The distance dG (u, v) between two vertices u, v ∈ V is the length of a shortest path from u to v in the graph G. If it is clear from the context, we omit the superscript G in the above notations and write N (v), deg(v), etc., instead. Every nonempty subset S ⊆ V (G) induces a subgraph G[S] with the vertex  set S and the edge set E(G[S]) := S2 ∩ E(G). Consequently, a graph H is an induced subgraph of G if H = G[V (H)]. Recall that H is a k-edge induced subgraph of G for k := |E(H)|. Again, let S be a set of vertices in G. Then S is a clique, if for every u, v ∈ S we have either u = v or {u, v} ∈ E(G). On the other hand, the set S is an independent set in G, if {u, v} ∈ / E(G) for all u, v ∈ S. For every k ∈ N, there exists a constant Rk , known as the Ramsey number, such that every graph G with |V (G)| ≥ Rk has either a clique of size k or an independent set of size k. It is well-known that Rk < 22·k for every k ∈ N.

2.3

Relational structures and first-order logic

A vocabulary τ is a finite set of relation symbols. Each relation symbol has an arity. A structure A of vocabulary τ , or simply structure, consists of a nonempty set A called the universe, and an interpretation RA ⊆ Ar of each r-ary relation symbol R ∈ τ . For example, a graph G can be identified with a structure A(G) of vocabulary τgraph := {E} with symbol E such that A(G) := the binary relation V (G) and E A(G) := (u, v) {u, v} ∈ E(G) . The disjoint union of two τ -structures A1 and A2 is again a τ -structure, denoted by A1 ∪˙ A2 , whose universe is A1 ∪˙ A2 , and where for each relation symbol R ∈ τ we let RA1 ∪˙ A2 := RA1 ∪˙ RA2 . Let A be a structure of a vocabulary the Gaifman graph of A is  τ . Then G(A) := (V, E) with V := A and E := {a, b} a, b ∈ A, a 6= b, and for someR ∈ τ, and some tuple(a1 , . . . , ar ) ∈ RA , {a, b} ⊆ {a1 , . . . , ar } . Note any unary relation in A has no influence on E.  r ∈ N and a ∈ A. Then the r-neighborhood of a is NrA (a) := b ∈ Let A dG(A) (a, b) ≤ r . Moreover, the structure NrA (a) induced by the r-neighborhood of a has universe NrA (a), and symbol R ∈ τ the for each r-ary relation interpretation (a1 , . . . , ar ) ∈ RA a1 , . . . , ar ∈ NrA (a) . Formulas of first-order logic of vocabulary τ are built up from atomic formulas x = y and Rx1 . . . xr where x, y, x1 , . . . , xr are variables and R ∈ τ is of arity r, using the boolean connectives and existential and universal quantification. 2.4

Tree-width and local tree-width

We assume that the reader is familiar with the notion of tree-width tw(G) of a graph G. Recall that the tree-width tw(A) of a structure A is simply tw(G(A)), that is, the tree-width of the Gaifman graph of A. In fact, to understand most parts of our proofs and algorithms, it is sufficient to know that for every structure A we have tw(A) < |A|. Now we are ready to define  the local tree-width of a structure A. For every r ∈ N let ltw(A, r) := max tw NrA (a) a ∈ A . Let g : N × N → N be a function and p ∈ N. We say a structure A has local tree-width bounded by g with respect to p if ltw(A, r) ≤ g(r, p) for every r ∈ N. This slightly generalizes the usual notion of local tree-width bounded by a unary function [13].

3

Some easy positive instances

In this section, let k ∈ N and G = (V, E) be a graph. Definition 1. G contains a k-independent-set-matching structure on vertices u1 , . . . , uk , v1 , . . . , vk if u1 , . . . , uk , v1 , . . . , vk are pairwise distinct; for every i, j ∈ [k] we have {ui , vj } ∈ E if and only if i = j; and {u1 , . . . , uk } is an independent set in G. Lemma 1. Every graph containing a k-independent-set-matching structure has a k-edge induced subgraph.

Proof: The case for k = 0 is trivially true. So assume k ≥ 1 and G contains a k-independent-set-matching structure on the vertices u1 , . . . , uk , v1 , . . . , vk .  ′ We choose the maximum k ≤ k such that ℓ := E G[{v1 , . . . , vk′ }] ≤ k. If   k ′ = k, then G[V ′ ] with V ′ := u1 , . . . , uk−ℓ ∪ v1 , . . . , vk is a k-edge induced  subgraph of G. Otherwise, k ′ < k. In particular, E G[{v1 , . . . , vk′ , vk′ +1 }] > k. As vk′ +1 can contribute at most k ′ many new edges, we have ℓ + k ′ > k, i.e.,   ′ ′ ′ k − ℓ < k . Then G[V ] with V := u1 , . . . , uk−ℓ ∪ v1 , . . . , vk′ is a k-edge induced subgraph of G. ⊓ ⊔ Definition 2. G contains a k-clique-matching structure on vertices u1 , . . . , uk , v1 , . . . , vk if u1 , . . . , uk , v1 , . . . , vk are pairwise distinct; for every i, j ∈ [k] we have {ui , vj } ∈ E if and only if i = j; and {u1 , . . . , uk } is a clique in G. Lemma 2. If G contains a k-clique-matching structure, then G has a k-edge induced subgraph. Proof: The  cases for k ≤ 2 are trivial. So we consider k ≥ 3. Let k0 be maximum with k20 ≤ k and set r := k − k20 . It is easy to verify that k ≥ k0 + r by k ≥ 3 and k0 > r. Now assume G contains a k-clique-matching-structure on the ′ vertices ≤ r such that u1 , . . . , uk , v1 , . .. , vk . Then, we choose the maximum r  ′ ′ ′ ℓ := E G[{v1 , . . . , vr′ }] ≤ r. If r = r, then G[V ] with V := v1 , . . . , vr ∪  subgraph of G. Otherwise, u1 , . . . , ur−ℓ , ur+1 , . . . , uk0 +ℓ is a k-edge induced  ′ ′ r < r and by the maximality of r we have E G[{v1 , . . . , vr′ , vr′ +1 }] > r. As ′ vr′ +1 can add at most r′ many new edges,  we have ℓ + r > r, or equivalently r − ′ ′ ′ ′ ℓ < r . It follows that G[V ] with V := v1 , . . . , vr ∪ u1 , . . . , ur−ℓ , ur′ +1 , . . . , ur′ +k0 −r+ℓ has exactly k edges. ⊓ ⊔

Definition 3. We say that G contains a k-apex structure on v0 , A and B if (A1) (A2) (A3) (A4)

A, B ⊆ V are disjoint with |A| ≥ k and |B| ≥ Rk , v0 ∈ V ; A is a clique in G; {u, v0 } ∈ E for every u ∈ A and {v, v0 } ∈ / E for every v ∈ B; {u, v} ∈ E for every u ∈ A and v ∈ B.

Lemma 3. If G contains a k-apex structure, then it has a k-edge induced subgraph. Proof: The case for k ≤ 1 is trivially true. So let k ≥ 2. Moreover, let v0 , A, B be as stated in Definition 3. Since |B| ≥ Rk , G[B] contains either a clique of size ′ k or an independent set of size k. If G[B] contains an independent  ′ set B ⊆ B ′ with |B | = k. Then for every u ∈ A the induced subgraph G B ∪ {u} has exactly k edges by (A4). Now assume that there is a clique B ′ in G[B] of size k. Observe by (A3) and k ≥ 2, we / (A ∪ B ′ ). Furthermore, it is easy to  have v0 ∈ k0 see that we can write k = 2 + r for some appropriate k ≥ k0 ≥ r. We select ′ arbitrary subsets A′ ⊆ A and B ′′ ⊆ |A′ | =r and |B ′′ | = k0 − r. Then it  B′ with ′′ is straightforward to check that G A ∪ B ∪ {v0 } has exactly k edges. ⊓ ⊔

Lemma 4. Assume there exists three disjoint cliques S1 , S2 , S3 in G, all of size k; and there are no edges between any distinct Si and Sj . Then G has a k-edge induced subgraph. It is easy to see that Lemma 4 is a direct consequence of Gauss’ famous Eureka Theorem [1].  Theorem 1. For every k ∈ N there exist k0 , k1 , k2 ∈ N such that k = k20 +   k2 k1 2 + 2 .

Lemma 5. Let k ∈ N+ and G = (V, E) be a graph without isolated vertices. If G contains an independent set of size (k − 1)2 + 1, then it has a k-edge induced subgraph. To prove the above lemma, we need some further preparation. + Lemma 6. Let m, n ∈ N and A, B ⊆ V be disjoint. If for every u ∈ A we have N (u) ∩ B ≥ 1 and |A| > (m − 1)(n − 1), then

(i) either there are m vertices u1 , . . . , um in A, and a vertex v in B with {ui , v} ∈ E for every i ∈ [m], (ii) or there are n vertices u1 , . . . , un in A and n vertices v1 , . . . , vn in B such that for all i, j ∈ [n] we have {ui , vj } ∈ E if and only if i = j. Proof: [of Lemma 5] Let S ⊆ V be an independent set in G with |S| > (k − 1)2 . Since G has no isolated vertex, |N (u) ∩ N (S)| ≥ 1 for every u ∈ S. So we can apply Lemma 6 on A ← S, B ← N (S), m ← k, and n ← k. If (i) holds, then we have an induced k-star of exactly k edges. Otherwise, we have (ii). Hence, there exist vertices u1 , . . . , uk ∈ S and v1 , . . . , vk ∈ N (S) such that G contains a k-independent-set-matching structure on those vertices. The result follows from Lemma 1. ⊓ ⊔ Definition 4. Let d ∈ N. We define  G V[1,d] := v ∈ V | 1 ≤ deg(v) ≤ d . It is well-known that if a graph contains many small-degree vertices, then it has a large independent set. As a result, the following is an easy consequence of Lemma 5. G Lemma 7. Let d, k ∈ N+ . If V[1,d] > (d+1)·(k −1)2 , then G contains a k-edge induced subgraph. 3.1

A further combinatorial lemma

For later purpose, we need a generalization of Lemma 6. + Lemma 8. Let in the graph G. If for m, n, p ∈ N and A, B ⊆ V be disjoint every u ∈ A, N (u) ∩ B ≥ p and |A| > (m − 1)(n − 1)p , then

(i) either there are m vertices u1 , . . . , um in A and p vertices v1 , . . . , vp in B with {ui , vj } ∈ E for every i ∈ [m] and j ∈ [p], (ii) or there are n vertices u1 , . . . , un in A and n vertices v1 , . . . , vn in B such that for all i, j ∈ [n] we have {ui , vj } ∈ E if and only if i = j.

4

Easy instances by model-checking

In this section we show the fixed-parameter tractability of p-Edge-InducedSubgraph on some restricted classes of graphs via the model-checking problem for first-order logic. The following is a generalization of a well-known result due to Frick and Grohe [13]. Theorem 2. For every computable function g : N × N → N the problem p-Mc-Ltwg -FO Instance: A structure A, p ∈ N and an FO-sentence ϕ such that A has local tree-width bounded by g with respect to p. Parameter: p + |ϕ|. Problem: Decide whether A |= ϕ. is fixed-parameter tractable. Definition 5. Let d ∈ N and G = (V, E) be a graph. If deg(v) ≤ d or deg(v) ≥ |V | − 1 − d for every v ∈ V , then the graph G is d-degree-extreme. Proposition 1. Let D : N → N be a computable function. Then the problem Instance: A graph G and k ∈ N such that G is D(k)-degreeextreme. Parameter: k. Problem: Decide whether G contains a k-edge induced subgraph. is fixed-parameter tractable. Definition 6. Let d, b ∈ N. Then (G, V1 , V2 , B) is a (d, b)-bridge (of the two degree-extreme graphs) if : (B1) V = V1 ∪ V2 for some disjoint V1 and V2 . (B2) G[V1 ] and G[V2 ] are both d-degree-extreme. (B3) B is a subset of V with |B| = b such that for every edge {u, v} with u ∈ V1 and v ∈ V2 we have either u ∈ B or v ∈ B. Similar to Proposition 1, we can prove: Proposition 2. Let D : N → N be a computable function. Then the problem Instance: A graph G = (V, E), V1 , V2 , B ⊆ V and k ∈ N such that (G, V1 , V2 , B) is a (D(k), |B|)-bridge. Parameter: k + |B|. Problem: Decide whether G contains a k-edge induced subgraph. is fixed-parameter tractable.

5

The algorithm

The main component of our fpt-algorithm for p-Edge-Induced-Subgraph is the following procedure that either already solves the problem or decomposes the given graph into potentially a bridge of two large degree-extreme graphs (cf. Definition 6). For every k ∈ N we let pk := 22·k (> Rk ). Lemma 9. For every computable function D : N → N there is an fpt-algorithm AD such that for every graph G = (V, E) and every k ∈ N exactly one of following conditions is satisfied. (S1) G is D(k)-degree-extreme and AD correctly decides whether G contains a k-edge induced subgraph. (S2) G is not D(k)-degree-extreme and AD correctly outputs that G contains a k-edge induced subgraph. (S3) G is not D(k)-degree-extreme and AD outputs three subsets V1 , V2 , B ⊆ V such that (S3.1) V = V1 ∪˙ V2 with |V1 | > D(k) and |V2 | > D(k) + 1; (S3.2) every edge between V1 and V2 in G has one vertex in B and |B| ≤ (pk − 1)pk +1 + (pk − 1)2 . Proof: Let G = (V, E) be a graph and k ∈ N. If G is D(k)-degree-extreme, then we apply Proposition 1 to achieve (S1). Otherwise let v0 ∈ V be a vertex with D(k) < deg(v0 ) < |V | − 1 − D(k).

(1)

Then we set V1 := N (v0 ) and V2 := V \ V1 . By (1) it holds that |V1 | > n D(k) and |V2 | = |V | − |V1 | = |V | − deg(v0 ) > D(k) + 1, i.e., (S3.1). Let W1 := u ∈ o V1 N (u) ∩ V2 ≥ pk and W2 := V1 \ W1 .

Claim 1. If |W1 | > (pk − 1)pk +1 , then G contains a k-edge induced subgraph. Proof of the claim. We apply Lemma 8 on A ← W1 , B ← V2 , m ← pk , n ← pk , and p ← pk . So there are pk vertices u1 , . . . , uqk in W1 and pk vertices v1 , . . . , vpk in V2 such that (i) either {ui , vj } ∈ E for every i, j ∈ [pk ], (ii) or for all i, j ∈ [pk ] we have {ui , vj } ∈ E if and only if i = j.

Recall pk > Rk , so there is a subset S ⊆ {u1 , . . . , upk } such that S is either an independent set or a clique. If S is an independent set, then G S ∪ {v0 } has exactly k edges. So suppose S is a clique. Assume that (i) is true, then G contains a k-apex structure on v0 , S, {v1 , . . . , vpk }. Hence, Lemma 3 implies the claim. Otherwise (ii) holds. And say S = {ui1 , . . . , uik }. Then the graph G contains a k-clique-matching structure on ui1 , . . . , uik , v1 , . . . , vk . The result follows from Lemma 2. ⊣ Claim 2. If N (W2 )∩V2 > (pk −1)2 , then G contains a k-edge induced subgraph. Proof of the claim. It is easy to verify that we can apply Lemma 6 on A ← N (W2 ) ∩ V2 , B ← W2 , m ← pk , and n ← pk . So,

(i) either there are pk vertices u1 , . . . , uqk in N (W2 ) ∩ V2 and a vertex v in W2 such that {ui , v} ∈ E for every i ∈ [pk ], (ii) or there are pk vertices u1 , . . . , upk in N (W2 ) ∩ V2 and pk vertices v1 , . . . , vpk in W2 such that for all i, j ∈ [pk ] we have {ui , vj } ∈ E if and only if i = j. But (i) contradicts our definition of W2 , i.e., for every u ∈ W2 we have N (u) ∩ V2 < pk , therefore (ii) must hold. Recall pk > Rk , hence G {v1 , . . . , vpk } contains either a clique of size of k or an independent set of size k. Without loss of generality, let {v1 , . . . , vk } ⊆ W2 ⊆ V1 be a clique or an independentset. For the independent set case, as v0 ∈ / V1 , then G {v0 , v1 , . . . , vk } is a kinduced subgraph. For the clique case, G contains a k-clique-matching structure on u1 , . . . , uk , v1 , . . . , vk . We are ⊣  done by Lemma 2. Let B := W1 ∪ N (W2 )∩V2 . If |B| > (pk −1)pk +1 +(pk −1)2 , then, by Claim 1 and Claim 2, the graph G contains a k-edge induced subgraph, and (S2) follows. Otherwise |B| ≤ (pk − 1)pk +1 + (pk − 1)2 . Observe that every edge between V1 and V2 has at least one vertex in B. Thus, we achieve (S3) by outputting (V1 , V2 , B). ⊓ ⊔ Finally we are ready to present our fpt-algorithm for p-Edge-InducedSubgraph. Theorem 3. p-Edge-Induced-Subgraph is fixed-parameter tractable. Proof: We define a computable function D0 : N → N by  2 D0 (k) := 2 · (pk − 1)pk +1 + (pk − 1)2 + 22·((k−1) +1) .

(2)

Then let AD0 be the algorithm as stated in Lemma 9 for the function D0 . Let (G, k) with G = (V, E) be an instance of p-Edge-Induced-Subgraph. First, we remove all the isolated vertices in G. For simplicity, the resulting graph is denoted by G again. Then, we simulate the algorithm AD0 on (G, k). If the result is either (S1) or (S2) in Lemma 9, we already get the correct answer. Otherwise, AD0 outputs three subsets V1 , V2 , B ⊆ V satisfying (S3.1) and (S3.2). If G[V1 ] and G[V2 ] are both D0 (k)-degree-extreme, then (G, V1 , V2 , B) is a (D0 (k), |B|)-bridge with |B| bounded by an appropriate computable function of k. The fixed-parameter tractability of whether G contains a k-edge induced subgraph follows from Proposition 2. Otherwise, either G[V1 ] or G[V2 ] is not D0 (k)-degree-extreme. We assume that G[V1 ] is not D0 (k)-degree-extreme. (The case for G[V2 ] is symmetric.) Then we simulate the algorithm AD0 on (G[V1 ], k). Observe that the result cannot be (S1). If the output is (S2), since G[V1 ] is an induced subgraph of G, we conclude that G has an induced subgraph of exactly k edges. Now we are left with case (S3). In particular, there are subsets V11 , V12 , B1 ⊆ V1 such that the corresponding properties of (S3.1) and (S3.2) are satisfied. Let U1 := V11 \ (B ∪ B1 ), U2 := V12 \ (B ∪ B1 ), and U3 := V2 \ (B ∪ B1 ). Observe that in G if we remove the vertex set B, then there is no edge left between V1 and V2 . Similarly, if we remove the vertex set B1 , every edge between V11 and V12

is destroyed. Thus, by (S3.2), in the original graph G, there is no edge between each pair of U1 , U2 and U3 . Moreover by (S3.1) and (S3.2) for every i ∈ [3]  2 |Ui | > D0 (k) − 2 · (pk − 1)pk +1 + (pk − 1)2 = 22·((k−1) +1) > R(k−1)2 +1 , where the equality is by (2). We use Ramsey’s Theorem again. If there is an independent set of size (k − 1)2 + 1 in one of the U1 , U2 and U3 , as G has no isolated vertex, then G contains a k-edge induced subgraph by Lemma 5. Otherwise every Ui contains a clique of size (k − 1)2 + 1 ≥ k. As we have seen that there is no edge between U1 , U2 and U3 in G, Lemma 4 implies that G contains an induced subgraph of exactly k edges. ⊓ ⊔

6

Counting k-edge induced subgraphs

The most natural counting version of p-Edge-Induced-Subgraph is: p-#Edge-Induced-Subgraph Instance: A graph G and k ∈ N. Parameter: k. Problem: Compute the number of k-edge induced subgraphs in G. In fact, the hardness of p-#Edge-Induced-Subgraph is rather easy to show. We observe that the vertex set of every induced subgraph without any edge is an independent set, and vice versa. Hence the first slice of p-#EdgeInduced-Subgraph, i.e., counting the number of 0-edge induced subgraphs is exactly the classical problem #Independent-Set of counting the number of independent sets in a given graph. Recall that #Independent-Set is #Phard [16]. Hence: Theorem 4. Assume #P 6= P. Then p-#Edge-Induced-Subgraph is not fixed-parameter tractable. One might attribute the above hardness result to the fact that we allow induced subgraphs to have isolated vertices. Note these isolated vertices play no role in the decision problem p-Edge-Induced-Subgraph. Therefore, it also makes sense to consider: p-#Edge-Induced-Subgraph∗ Instance: A graph G and k ∈ N. Parameter: k. Problem: Compute the number of k-edge induced subgraphs without isolated vertices in G. Then we can show: Theorem 5. p-#Edge-Induced-Subgraph∗ is hard for #W[1].

Acknowledgement We thank Leizhen Cai for bringing the problem p-Edge-Induced-Subgraph to our attention, and J¨org Flum for comments on earlier versions of this paper. This research has been partly supported by the National Nature Science Foundation of China (60970011, 61033002).

References 1. G. Andrews. Eureka! num = ∆+∆+∆. Journal of Number Theory, 23(3):285–293, 1986. 2. H.L. Bodlaender, L. Cai, J. Chen, M.R. Fellows, J.A. Telle, and D. Marx. Open problems in parameterized and exact computation - IWPEC 2006. Technical Report UU-CS-2006-052, Department of Information and Computing Sciences, Utrecht University, 2006. 3. L. Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58(4):171–176, 1996. 4. L. Cai. Private communication, 2008. 5. L. Cai, S.M. Chan, and S.O. Chan. Random separation: A new method for solving fixed-cardinality optimization problems. In Proceedings of the second International Workshop on Parameterized and Exact Computation (IWPEC’06), Lecture Notes in Computer Science 4169, pages 239–250. Springer, 2006. 6. Y. Chen and J. Flum. On parameterized path and chordless path problems. In Proceedings of 22nd Annual IEEE Conference on Computational Complexity (CCC 2007), pages 250–263. IEEE Computer Society, 2007. 7. Y. Chen, M. Thurley, and M. Weyer. Understanding the complexity of induced subgraph isomorphisms. In Proceedings of the 35th International Colloquium on Automata Languages and Programming (ICALP’08), pages 587–596. Springer, 2008. 8. B. Courcelle. Graph rewriting: An algebraic and logic approach. In J. Van Leeuwen, editor, Handbook of Theoretical Computer Science, pages 192–242, Amsterdam, The Netherlands, 1990. Elsevier Science Publishers. 9. R.G. Downey and M.R. Fellows. Parameterized Complexity. Springer-Verlag, 1999. 10. Z. Dvorak, D. Kr´ al, and R. Thomas. Deciding first-order properties for sparse graphs. In Proceedins of the 51th Annual IEEE Symposium on Foundations of Computer Science (FOCS’10), pages 133–142. IEEE Computer Society, 2010. 11. J. Flum and M. Grohe. The parameterized complexity of counting problems. SIAM Journal on Computing, 33(4):892–922, 2004. 12. J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. 13. M. Frick and M. Grohe. Deciding first-order properties of locally tree-decomposable structures. Journal of ACM, 48(6):1184–1206, 2001. 14. S. Khot and V. Raman. Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science, 289(2):997–1008, 2002. 15. D. Seese. Linear time computable problems and first-order descriptions. Mathematical Structures in Computer Science, 6(6):505–526, 1996. 16. L. G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410–421, 1979.

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